Anonymous users can create content if you give them permissions in the back end. Go to yoursite.com/admin/user/permissions and make scroll down to the "Node Module". Make sure you have "access content" and any relevant "create **foo** content" boxes checked (where **foo** is the content type you want them to be able to create) for anonymous users.
This will grant anonymous users permission to create content of type **foo**, and view any content created.
Your "Access Denied" problem may be caused by anonymous users being able to create content, but not access it. This would be the case if you have granted "create **foo** content" permissions to the anonymous user, but have not given them "access content" permissions. Can you see the content while you are logged in as an admin?